REAL AND MONEY CONSUMPTION AS FUNCTIONS OF MONEY INCOME

ROGER NILS FOLSOW NAVAL POSTGRADUATE SCHOOL

Most presentations of the “standard” Keynesian macroeconomic model include a consumption function in which real (deflated1) consumption expenditure is a function of real income. Such a function assumes that a rise in output prices (and thus in the money value of income) does not affect aggregate real consumer demand, unless the price rise changes aggregate real output and real national income.2 Thus validity for a consumption function written in real terms requires that money illusion not affect aggregate real consumption, and that changes in the structure of relative prices either not accompany price level changes or else not affect aggregate real consumption (either through substitution or income effects).

Of course, a consumption function written in real terms can be restated in money terms, without abandoning the basic hypothesis that real consumption is independent of the price leveLS Although such a conversion is unquestionably valid formally, what are its implications? To answer that question, this note derives and discusses some properties of two alternative formulations of the consumption relationship: real con- sumption as a function of money income, and mon consumption as a function of

the price level. It condudes by discussing the graphical implications of these results.

GENERAL CONSUMPTION RELATIONSHIPS

money income, without and also with a direct depen 7 ence of real consumption upon

In general, real consumption as a function of money may be written

and money consumption as a function of money income may be written

where t and q are real consumption and income respecbvely, p is the price level, and p and 4 are independent of each other. To allow for the possibility that aggregate

consumption behavior may be affected by prices, let >-o indicate the effect (if

any) of prices upon consumption behavior. *Lieutenant, U.S. Naval Reserve. The views expressed do not necessarily reflect the opinion of

the US. Government, the Navy, or the Postgraduate School. Although fully responsible for any debcienaes, the author is indebted to Ann Arlene Folsom,

Fem C Horton, Paul E. Roberts, Jerry L. Dake, Henry C. Durham, and Frank I. Jewett for useful discussions and criticisms of preceding drafts. He is particularly indebted to Thomas H. Hibbard for suggesting the comparison between real and money consumption, and to Carl R. Jones for suggesting the use of elasticities. A longer version of this note, including an introductory discussion using elementuy linear functions and their graphs, is available under the same title, as Naval Postgraduate School Technical R

’Whether deflation is by an index x h e level of output prices (the usual practice in recent writings) or input prices (“wages” in Keynes’ analysis) is basically immaterial to the analysis presented here, provided that consumption and income are deflated by the same prices. For con- venience the discussion assumes deflation by output prices.

*In a full model containing monetary and production sectors, other relationships will make real national income and the price level dependent upon each other. But this note is concerned only with a ceterir paribus partial equilibrium analysis of the consumption function alone, not with a general equilibrium analysis of a complete macroeconomic model.

‘Fmco Modigliani’s classic description of the Keynesian system 131 used a consumption func- tion in money terms but assumed that real consumption was independent of prices, for example that consumers were not nffected by money illusion.

96

(1) c = j (pq )

(2) Pc = g ( P d

ap<

rt NF’S-62FL903 IA.

FOLSOM: CONSUMPTION FUNCTION 91

Then

ac ac ap a4

dc = - dp + - dq

Therefore, the slopes of real and money consumption with respect to money income ate

(3)

and

(4)

The elasticities of real and money consumption with respect to money income are

(3’)

(4‘)

Since the effect of prices upon the relationship between real consumption and money income differs from its effect upon the relationship between money consumption and money income, the difference in the linearity of these two relationshi s in the presence of price effects might furnish the basis for an econometric test o P the resence of money illusion or other price influences upon consumption.

If real output varies while prices are constant, dp = 0 so that the presence of price effects is irrelevant. Whenever prices are constant, the relationship between real con- sumption and money income is

[2, a. sf

and the relationship between money consumption and money income is

The slo e of money consumption as a function of money income is identical to the slope o P real consumption .as a function of real income, if prices are constant.

If prices vary while real output is constant, dq = 0 SO that the relationship between red consumption and money income is

98 WESTERN ECONOMIC JOURNAL

with a price effect, and is zero in the absence of a price ef€ect. The relationship between money consumption and money income is

with a price effect, and simply equals c / q in the absence of a price effect. If real output is constant and there is no price ef€ect, money consumption as a linear function of money income lies along a ray through the origin, with a slope e q d i n g the average propensity to consume. If real output is constant and higher prices decrease consumption, for example if consumers suffer from money illusion in which they notice a price rise but do not notice the accompanying increase in money income,

- < o and money consumption as a function of money income will be less steep than

%e average propensity to consume. If real output is constant and higher prices increase consumption, for example if consumers suffer from a money illusion in which they do not notice a price rise but do notice the accompanying increase in money income,

- > o and money consumption as a function of moneyincome will be steeper than

the average propensiq to consume.

ac

ac aP

CONDIT IONS FOR GRAPHING CONSUMPTION O N MONEY INCOME

If prices are constant while output varies, or if prices vary while output is constant, there is no difficulty in graphing either real or money consumption as a function of money income. Bqt of course the slope of either graph will differ, depending on whether the cetetir p.ribus assumption is constant prices or constant people.

More generally neither prices nor output will remain constant; both will vary simultaneously. In this case, a valid graph of either real or money consumption as a function of money income, with consumption on one and money income on the other of two axes, is not possible unless certain conditions are satisfied. There are too many independent variables - two ( p and ) instead of one - with each of these inde- pendent variables affecting the depen 1 ent variable differently. Even though neither prices nor ou ut remain constant, however, such a valid graph is possible, if con-

Thus if both output and prices are allowed to vary, the formal condition for a valid two-dimensional graph of real consumption on one axis as a function of money income on the other axis is that equation (3b) equals (3a) :

sumption is a# ected equally by increased output q or by a rise in. prices p .

so that the slope of real consumption as a function of money income decreases in steepness as money income increases, whether due to increased real income or to a rise in price levels (if ar/ap and are constant). In this special case the money income elasticity of real consumption expenditure is

Thus the condition is that the price level elasticity must equal the real income elasticity of real consumption expenditure. Money illusion or other price etfects must be “com- plete,” so that a rise in prices has the same percentage &ect as a rise in real income.

Similarly, if both output and prices are allowed to vary, the formal condition for a

FOLSOM: CONSUMPTION FUNCTION 99

valid two-dimensional graph of money consumption on one axis as a function of money income on the other axis is that equation (4b) equals (4a) :

so that the slope of money consumption as a function of money income is constant (if the marginal propensity to consume is constant). In this special case the money income elasticity of money consumption expenditure is

Thus this condition is that one plus the price level elasticity must equal the real income elasticity of real consumption expenditure. (For example, suppose that prices

have no effect upon aggregate real consumption,--= 0, and that the marginal equals

the average propensity to consume,---. = - so that in a linear consumption function

the constant is zero.) In short, if both output and prices are allowed to vary, then graphing either real

or money consumption on one axis as a function of money income on the other axis requires rather exacting assumptions.4

The difficulty of graphing consumption as a function of money income is particularly unfor- tunate, because it causes simple graphic analyses of the tax side of fiscal policy to be generally incorrect. Consider a model in which real consumption is a function of real national income after real tax collections (the real output tramfared to govcmment via taxes), money tax collections arc a function either of money income or of money consumption, and output and prices vary simultaneously. In such a model, analyzing tax policy by using a simple graph of consum tion as B function of money income before taxes (so that tax increases shift this consumption function down) is tempting, but generally wrong because the consumption graph is invalid. For an analysis of tax policy that avoids this problem only by assuming that output and prices do not vary simul- taneously, see Brown [13 and Ritter 141.

ac

aP ac c a4 4

REFERENCES

1. E. Cary Brown, “Analysis of Consumption Taxes in Terms of the Theory of Income Determi- nation.” Am. Econ. Rev., March 1950.40, 7489.

2. Franklin M. Fisher, T b e Identification Problem in Econometrics. New York 1966. 3. Franc0 Modigliani, “Liquidity Preference and the Theory of Interest and Money,” Econome:rku,

Jan. 1944, 12, 45-88; reprinted in Friedrich A. Lutz and Lloyd W. Mints (A Committee of the American Economic Assn.), Readings in Monehzry Tbeory. Homewood 1951.

4. Lawrence S. Ritter, “Consumption Taxes and Income Determination: Comment,” Am. Econ. Rm., Mpmh 1951,41, 191-93.

Tbe Economics Institute for foreign students of economics who are beginning graduate work in the United States will be held under the sponsorship of the American Economic Association from June 18 to August 20, 1969 at the University of Colorado. Information and application forms may be obtained from the Director of the Economics Institute, University of Colorado, Boulder, Colo- rado 80302, USA.